A note on the differential spectrum of a class of locally APN functions

TL;DR

This work addresses the differential properties of the binomial function $f_u(x)=x^{\frac{p^n+3}{2}}+ux^2$ over ${\mathbb{F}}_{p^n}$ with $u=\pm1$, building on prior results for $u$ excluding $0,\pm1$. The authors connect the differential spectrum to quadratic character sums, notably introducing and employing the sum $\lambda_{p,n}=\sum_{x\in\mathbb{F}_{p^n}}\chi(x(x^2-2x-1))$ and the parity-dependent character $\chi(2)$, to derive explicit spectrum formulas. For the case $f_1(x)=x^{\frac{p^n+3}{2}}+x^2$ with $p^n\equiv3\pmod4$, they prove the differential uniformity $\Delta_{f_1}=(p^n+1)/4$, compute $N_4$, and present the full differential spectrum depending on $\lambda_{p,n}$ and $\chi(2)$, with notable special cases reducing to PN, APN, or locally APN behavior. The results yield a detailed, sum-based characterization of the spectrum, providing both exact expressions and practical bounds, and suggest broader applicability to the differential analysis of other cryptographic functions via quadratic character sums.

Abstract

Let $\gf_{p^n}$ denote the finite field containing $p^n$ elements, where $n$ is a positive integer and $p$ is a prime. The function $f_u(x)=x^{\frac{p^n+3}{2}}+ux^2$ over $\gf_{p^n}[x]$ with $u\in\gf_{p^n}\setminus\{0,\pm1\}$ was recently studied by Budaghyan and Pal in \cite{Budaghyan2024ArithmetizationorientedAP}, whose differential uniformity is at most $5$ when $p^n\equiv3~(mod~4)$. In this paper, we study the differential uniformity and the differential spectrum of $f_u$ for $u=\pm1$. We first give some properties of the differential spectrum of any cryptographic function. Moreover, by solving some systems of equations over finite fields, we express the differential spectrum of $f_{\pm1}$ in terms of the quadratic character sums.

Theorems & Definitions (27)

• Definition 1
• Lemma 1
• Lemma 2
• Example 1
• Example 2
• Lemma 3
• proof
• Definition 2
• Theorem 1
• proof